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arXiv · 2610.31308

The Bayesian Learning Rule Beyond KL Geometry

Sophia Sklaviadis·Wu Lin·Andre Martins·Mario Figueiredo·Thomas Moellenhoff·Emtiyaz Khan
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Abstract

The Bayesian learning rule (BLR) exploits the relation between Bayesian updating and the Kullback-Leibler divergence (KLD) to derive efficient additive updates for exponential family variational posteriors. The KL geometry is powerful but restrictive: combined with posterior families with parameter-dependent support it can lead to undefined objectives. We ask which non-KL geometries preserve the additive BLR computation. Previous works that use $f$-divergences suggest that this is only possible with the KLD. Here, we show that this is not true. We show that additive information-geometric optimization algorithms can be derived for the class of mixed-space Bregman or Fenchel-Young discrepancies. These discrepancies induce generalized maximum-entropy families, dual coordinates, and generalized-BLR canonical gradient updates, recovering the classical BLR. The generalized BLR leads to new algorithms that \textit{cannot} be derived from the KL-based BLR. For the Tsallis negentropy with quadratic statistics, the generalized BLR yields bounded-support and heavy-tailed $q$-Gaussian posteriors and an easy Newton-like implementation. Experiments up to CIFAR-100 illustrate controllable support and tail behavior and competitive performance.

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Sophia Sklaviadis, Wu Lin, Andre Martins, Mario Figueiredo, Thomas Moellenhoff, Emtiyaz Khan. 2026-10-22. The Bayesian Learning Rule Beyond KL Geometry. https://arxiv.org/abs/2610.31308

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